A few days ago, Natalie Wolchover wrote an article about two famous Gödel's theorems. What is cool is that her text isn't just "some popular article about mathematics". It is really a basically full proof of the theorems translated into plain English – without sacrificing the substance.
A big part of her presentation is the translation (Gödel's numbering) of sequences of characters (strings: propositions, proofs...) into unique integer codes. The number of possible strings (or computer files) is countable which means that they may be identified with positive integers in a one-to-one fashion. You could fine-tune a code that translates posssible 16 characters into 0-F (hexadecimal digits) and then add "1" in front of the sequence, to get a hexadecimal number. Well, all of them would start with "1" (which means that not all possible integer codes are used, the map isn't quite one-to-one) but the problem could be fixed by changing the rules a little bit.
Instead, for historical reasons, she picks a perhaps even more elegant translation of final strings into integers: she understands the codes 0-15 as exponents above (increasingly large) primes. Again, it's not quite one-to-one but the problem may be fixed.
Then she discusses the liar's-paradox-like propositions and their codes which are inserted into the propositions themselves. The liar's paradox is when a person says "I am lying" which cannot be true but it cannot be false, either. A system of axioms that could prove or disprove every meaningful statement would behave like the liar's paradox, she repeats Gödel's proof in a more comprehensible way. The underlying idea is the liar's paradox but the individual steps must be done using Gödel's numbering technology and with all the caution "what we precisely mean". It follows that there exists a statement that can be neither proven nor disproven, otherwise the axiomatic system would suffer from a liar's-paradox-style inconsistency.
As always, whenever the liar's paradox is cleverly used in the precise axiomatic argumentation, you may informally say whether the statement is right. The statement of the type "I cannot be proven using some rules of the game" is clearly right – because the system would otherwise be inconsistent and it rather clearly isn't – but the proof is a meta-proof which cannot be achieved using the original rules of the game. If you tried to fix the original rules of the game so that they would allow you to prove that this particular proposition is impossible to be disproven, you would obtain another system that would still be incomplete – although the new undecidable statement would be a bit different.
The second Gödel's theorem using similar games to prove that the consistency of a strong enough system of axioms is impossible to be proven using the system itself. If it could be proven, you could also prove that it is an inconsistent system! So all mathematical "theories of everything", as she playfully calls them, are both incomplete (some statements are neither provable nor disprovable) and self-doubting (the consistency is unprovable).
More sociologically, I think that hers is the format of articles that should be way more widespread – about mathematics, physics, and other fields. Comprehensible presentations that actually contain all the important beef, something that the original author wouldn't be ashamed of, something that could actually be considered an improved presentation that is publishable in the technical journals. Well, yes, cutting-edge theoretical physics would probably be harder simply because no science writers really understand the stuff.
But we are living in the world where no articles like that exist. Every popular science article is just brainwashing the readers. They are led to memorize some clichés using the buzzwords that were said to be cool. But none of the readers of these "articles about physics and other sciences" actually understands anything. They may either repeat something they have read verbatim; or they may create their own propositions that are complete noise – the correct let alone deep statements aren't overrepresented in any way.
In recent years, I was increasingly realizing the breathtaking stupidity of an overwhelming majority of the readers of popular science texts (and viewers of popular science programs) who think that they are intelligent but they are not. Something like 99% of the pompous readers of articles about physics – and, sadly, this set may also include some 80% of TRF readers – are just morons who are only capable of repeating some words from the latest science-like text they have seen. But they don't actually understand anything at all. It means that when they read another, completely wrong and stupid, article about some question which contradicts something they "learned" previously, they just overwrite their pre-existing "knowledge" by the new one. The reason for this dysfunctional flexibility is simple: it wasn't really "knowledge". It was just something they parroted and if it happened to be right, the correctness was nothing else than a lucky coincidence.
They have actually no tools to distinguish correct statements from the incorrect ones (and sometimes from the absolutely idiotic ones). After decades of "reading popular articles about science", they have made strictly zero progress in their understanding of physics. They have completely wasted decades of their lives and their pretended "informed layman's understanding" is a complete fabrication. Just like they have wasted a lot of time, so have I. I couldn't believe that such a huge fraction of the people can be so incredibly dense. But it is demonstrably the fact. That's why everytime some new imbecile writes an article how quantum mechanics doesn't work or how loop quantum gravity is the theory of everything (with claims that have been explained to be wrong many, many times and very, very clearly), tons of other imbeciles spam my mailbox demanding an excited reaction to a new copy of this garbage that is impossible to eliminate.
It is counterproductive to try to teach quantum mechanics let alone string theory to a truly large percentage of the people – because such an outcome is probably impossible – and it is extremely dangerous to assume that such an outcome can materialize in the future (or even that it is the reality now). Instead, it is extremely important to protect mathematics and science from this totally stupid majority, majority of brain-dead parrots who have no idea what they're talking about but who use the time that they have wasted by reading and parroting random words as an excuse for their misplaced self-confidence. Most of the stuff pretending to "teach advanced physics to the masses" isn't teaching anything. Instead, these things are "making morons more self-confident". The net contribution of this "popular science" to the society is almost certainly negative.